Friday, September 17, 2010

Why Music Works: Chapter Four

Root Motion and Voice Transformation Types in the Beta and Gamma Contextual Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Four:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect.

Chapter four is dedicated to examining how Beta Prime and Gamma Prime compare to Alpha, using the same musical proof formats.




Listen to Example 14

Example fourteen is the same as example eleven, but with the inflections necessary to put it into the Beta Prime system. Obviously, these are end-contextualized, and are comparing the progressive resolutions to their opposite regressive versions.

The first of these root motions - the one from vii(d5m7) to bIII(A5M7) - is labelled PA5, for Progressive Augmented Fifth (This could also conceivably be labelled a Pd4 for Progressive Diminished Fourth, but since falling fifths are the most natural progressive root motions, I've decided to stick with fifths here). [As you'll see, my terminology eventually evolved into calling these quadra-tones. - Geo] This is a surprising and uncanny effect, because of both the root motion and the structure of the target harmony. Immediately following that - into the vi(d5m7) - we get a Progressive Tritone, which is labelled Ptt in the analysis. This root motion can actually be found in the Alpha system when moving from IV(M7) to vii(d5m7), but since the original intent of this example was to put all seven harmonies of Alpha into normal progressive order, we have not seen that yet.

After these initial histrionics - a very nice resource to affect the listener into the realm of the uncanny - the rest of the progressive relationships are relatively normal.

Since the second system is essentially the first system in reverse, the strangeness occurs near the end there. When I created this example back in 2008, I had still not completely nailed down the way I wanted to treat the progressive and regressive augmented fifths, so there is an Rd4 in the analysis, but the final version will say Rqt there for regressive quadra-tone. Note that the Ptt in Alpha is IV(M7) progressing to vii(d5m7) while the Rtt there is moving from vii(d5m7) to IV(M7): Both root motions are by tritone, but one is progressive and the other is regressive. The same thing applies in PA5 versus RA5.


1. Overtone chord progressions imply falling fifths, hence, Progressive Augmented Fifth, Crosswise.

2. Overtone chord regressions imply rising fifths, hence, Regressive Augmented Fifth, Crosswise.

3. Progressive and regressive tritone root motions are perfectly usable in proper context.

4. Progressive and regressive augmented fifth root motions are perfectly usable in proper context.

5. Since augmented and diminished fifth root motions yield uncanny effects, employ accordingly.

6. The tonic minor/major seventh also evokes the uncanny.

7. The lowered mediant degree augmented/major seventh sonority also evokes the uncanny.

8. the harmonies and root motions possible in the Beta System provide new expressive resources.



Listen to Example 15

Example fifteen is the same as example twelve, but inflected to put it into Beta Prime.


1. Beta system .5P's and .5R's contain less dramatically uncanny effects than the P's and R's did.

2. This is because the descending and ascending thirds smooth out the tritones and quadra-tones.



Listen to Example 16

This is example thirteen inflected to put it into Beta Prime


1. Beta Prime SP's and SR's are again less uncanny than the system's P's and R's are.

In this case, all root motion and transformation is stepwise, hence the smoothness.



Listen to Example 17

Now we are ready to use example eleven/fourteen, example twelve/fifteen, and example thirteen/sixteen inflected into the Gamma System. By this point in creating these musical proofs, my terminology for the augmented fifth progressions and regressions had evolved to the point of referring to them as quadra-tones, which compares better to the tritone analysis symbols: Pqt and Rqt respectively, to fit in better with Ptt and Rtt. by this point, you should understand the analysis symbols well enough that the proofs become self-explanatory.


1. Gamma System progressions and regressions contain even more uncanny sounding effects.

2. The areas of alternating quadra-tone and tritone root motions sound particularly sinister.

Obviously, these are gnarly sonic resources.



Listen to Example 18

Here are the half-progressions and half-regressions in Gamma. Note that I neglected to parenthetically denote the diminished thirds for the vii(d3d5m7) sonorities here. They are still notated properly - and so they sound correct - so that is just an oversight on my part.


1. Gamma System .5P's and .5R's again contain fewer dramatically uncanny effects then the system's P's and R's do.

2. This is again, as before, because the .5P's and .5R's smooth out the tritones and quadra-tones.



Listen to Example 19


1. Gamma Prime SP's and SR's are again less dramatically uncanny sounding than the system's P's and R's are.

2. Again, the fact that all root motion and transformation is stepwise aids smoothness.

3. Nonetheless, the Gamma System is filled with unsettling harmonic effects, which is an effective and affective resource.

Now that we have seen and heard the basic resources of the normal diatonic Alpha, Beta and Gamma contextual systems, it is time to enter the chromatic realm with secondary dominants and secondary subdominants as derived from the Alpha system.


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